Questions: Find the 10th term of the arithmetic sequence 2, 6, 18, 54, ... a. 4,374 b. 13,122 c. 39,366 d. 118,098

Solution Steps

To find the 10th term of the given sequence, we first need to identify the type of sequence. The sequence provided is \(2, 6, 18, 54, \ldots\), which is a geometric sequence because each term is obtained by multiplying the previous term by a constant factor. We can determine the common ratio by dividing the second term by the first term. Once we have the common ratio, we can use the formula for the nth term of a geometric sequence, which is given by \(a_n = a_1 \times r^{(n-1)}\), where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number we want to find.

The given sequence is \(2, 6, 18, 54, \ldots\). This is a geometric sequence where each term is obtained by multiplying the previous term by a constant ratio.

To find the common ratio \(r\), we divide the second term by the first term: \[ r = \frac{6}{2} = 3 \]

The formula for the \(n\)th term of a geometric sequence is given by: \[ a_n = a_1 \times r^{(n-1)} \] where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number. For the 10th term (\(n = 10\)): \[ a_{10} = 2 \times 3^{(10-1)} = 2 \times 3^9 \]

Calculating \(3^9\): \[ 3^9 = 19683 \] Thus, \[ a_{10} = 2 \times 19683 = 39366 \]

Final Answer